Uniformization by square domains (Q524749)

The aim of the paper under review is an extremal problem for the normalized conformal mapping of an \(n\)-fold connected domain onto a square domain whose complementary components are single points or closed squares with sides parallel to the real or the imaginary axis. An existence proof for such a conformal mapping was given by \textit{H. Grötzsch} [Ber. Verh. Sächs. Akad. Leipzig 87, 319--324 (1935; Zbl 0014.16602)]. Let \( \Omega \) be an \(n\)-fold connected domain in the Riemann sphere \(\hat{\mathbb{C}} = \mathbb{C} \cup \infty \) and \(\mathcal{F} \) be the set of all conformal maps \( f : \Omega \to D = f(\Omega) \subseteq \hat{ \mathbb{C} } \) with the normalization \[ f(z) = z + \frac{a_1}{z} + \ldots \] for \( z \) near \(\infty\), then the extremal property is that the functional \[ f \in {\mathcal{F}} \to {\mathcal{S}} (f) := 2\pi \Re(a_1) + \sum_{j=1}^n (V_j^2 - A_j), \] where \(A_j\) denotes the area of the complementary components of \(D\) and \(V_j\) its vertical variation, has a unique minimizer \(f_0 \in \mathcal{F}\), which is the unique conformal map \(f_0: \Omega \to D \) onto a square domain \(D\). The proof is based on a theorem stating that for a square domain one has \( {\mathcal{S}} (f) \geq 0 \) with equality if and only if \(f\) is the identity.